 Research
 Open Access
Rangeazimuth decouple beamforming for frequency diverse array with Costassequence modulated frequency offsets
 Zhe Wang^{1},
 WenQin Wang^{1}Email authorView ORCID ID profile and
 Huaizong Shao^{1}
https://doi.org/10.1186/s1363401604223
© The Author(s) 2016
 Received: 20 June 2016
 Accepted: 5 November 2016
 Published: 23 November 2016
Abstract
Different from the phasedarray using the same carrier frequency for each transmit element, the frequency diverse array (FDA) uses a small frequency offset across the array elements to produce rangeangledependent transmit beampattern. FDA radar provides new application capabilities and potentials due to its rangedependent transmit array beampattern, but the FDA using linearly increasing frequency offsets will produce a range and angle coupled transmit beampattern. In order to decouple the rangeazimuth beampattern for FDA radar, this paper proposes a uniform linear array (ULA) FDA using Costassequence modulated frequency offsets to produce randomlike energy distribution in the transmit beampattern and thumbtack transmitreceive beampattern. In doing so, the range and angle of targets can be unambiguously estimated through matched filtering and subspace decomposition algorithms in the receiver signal processor. Moreover, randomlike energy distributed beampattern can also be utilized for low probability of intercept (LPI) radar applications. Numerical results show that the proposed scheme outperforms the standard FDA in focusing the transmit energy, especially in the range dimension.
Keywords
 Frequency diverse array (FDA) radar
 Frequency offset
 Rangedependent
 Decouple
 Costas sequence
1 Introduction
Active phasedarray has been widely adopted in many applications such as radar, electronic warfare, radio astronomy, etc., because it can steer the beam electronically with high effectiveness [1]. The offered directional gain is useful for detecting/tracking weak targets and suppressing sidelobe interferences in other directions [2]. Since phasedarray has rangeindependent transmit beampattern, the range and angle of targets cannot be unambiguously estimated from the beamforming output peaks. If we want to steer the array beams to multiple different range cells, multiple antennas or a multibeam antenna will be required. More importantly, controlling rangedependent energy distribution becomes an increasingly important requirement in many applications. While techniques exist in mitigating rangedependent interferences, e.g., spacetime adaptive processing [3], they generally require a high computational cost.
In order to overcome the disadvantage of phasedarray radar, the frequency diverse array (FDA) radar using a small frequency offset across the array elements was proposed in 2006 [4]. This steppedfrequency offset results in that the beam can scan the space in a periodic manner [5, 6]. Its beamforming focusing direction will change as a function of the range, angle, time, and even the frequency offset [7]. These characteristics contrast with the rangeindependent transmit beampattern in a phasearray radar. FDA radar is different from orthogonal frequency division multiplexing (OFDM) radar [8] and multipleinput multipleoutput (MIMO) radar [9, 10]. OFDM radar uses orthogonal subcarriers, but nonorthogonal carriers are employed in FDA radar. MIMO radar aims to provide noncoherent waveforms to obtain increased degrees of freedom (DOFs), whereas FDA radar transmits overlapping signals with closely spaced frequencies to provide additional functionality. FDA radar is also different from conventional frequency scanning radar using the frequency increments as a function of time for all the elements [11], but FDA frequency offsets are characterized by the element index [12]. Another similar concept is the timemodulation array [13], which weights each element using on/off switching operation. FDA was investigated in [14] as a rangedependent beam with applications in suppressing range ambiguous clutter. Secmen et al. [5] described the time and angle periodicity of FDA radiation pattern. Higgins and Blunt [15] explored rangeangle coupled beamforming in FDA. Additional studies to exploit FDA rangedependent beampattern characteristics were reported in [16, 17]. In fact, with the introduction of frequency increment, the FDA apparent angle will be different from its nominal beam scanning angle.
Due to its promising application potentials [18], FDA has sparked many interesting investigations [19–21]. Since FDA offers a rangeangledependent beampattern, it is of great importance as this provides a potential for rangeangle localization of targets, but the transmit beampattern of a standard FDA using linearly increasing frequency offsets is coupled in the rangeangle dimension. This limits its application for unambiguously estimating target parameters. To decouple the rangeangle coupling response of targets, a simple rangeazimuth localization of targets is proposed in [22] by adopting a uniform linear array (ULA) doublepulse FDA radar. This doublepulse FDA radar transmits two pulses with zero and nonzero frequency offsets, respectively. In [23], a subarraybased FDA is proposed for target rangeangle estimation. Furthermore, a transmit subaperturing is designed in [20] with convex optimization, so that the range and angle responses are decoupled and the equivalent transmit beam can be focused in a certain rangeangle sector to localize the targets.
In [19], a nonuniform linear array is adopted for the FDA. However, the transmitter and receiver must be placed accurately. Another nonuniform linear array for FDA is attempted to suppress/locate rangedependent interference/target in [12], but the carrier frequency and/or frequency increments cannot be altered in realtime because it requires relocating the elements mechanically. Logarithmically increasing frequency offsets [24] or timedependent frequency offsets [25] are also suggested to decouple the rangeangle beampattern, but they result in poor beamforming performance, especially in the range dimension. In fact, the best decoupling approach is to form a dotshaped beampattern rather than an “S”shaped beampattern. Such a dotshaped rangeangle beampattern is synthesized in [26] by a symmetrical FDA using multicarrier frequency offsets and convex optimization, and in [27] by the use of random frequency offsets. Nevertheless, these two schemes are very difficult to implement in practical array systems.
In this paper, we analyze the reason of FDA range and angle coupled transmit beampattern and thus propose the rangeazimuth decouple beamforming for FDA radar using Costassequence modulated frequency offsets. The rest of this paper is organized as follows. Section 2 provides a brief introduction to basic FDA radar scheme and motivation of this paper. Then, Section 3 proposes the rangeazimuth decoupling beamforming for the FDA using Costassequence modulated frequency offsets. Finally, numerical results are provided in Section 4 and concluding summaries are drawn in Section 5.
2 Basic FDA radar and motivation
with d being the element spacing.
where the common phase Φ _{0} is \(\Phi _{0}=2\pi f_{0}\left (t\frac {r}{c_{0}}\right)\).

1. Linearly increasing frequency offsets and nonuniform linear array

2. Uniform linear array and nonlinearly increasing frequency offsets
In this paper, we use Costassequence modulated frequency increments, namely, the second case.
3 FDA radar using Costassequence modulated frequency offsets
where a _{ i } is the ith element of the coding sequence. The remaining locations (where i+j>M) are left blank. This equation implies that the first row is formed by taking differences between adjacent elements in the coding sequence, the second row is formed by taking differences between nextadjacent elements, and so on.
As noted in Eq. (14), in the standard Costas sequence the subpulses corresponding to each element are not aligned. However, in the FDA the transmitted signals from all the elements should be aligned in timedomain; otherwise, the beampattern of the whole array will be decided mainly by some particular elements for a given instant. To avoid this problem, we use the Costassequence modulated frequency offsets in a timealigned way. Taking the Costas sequence illustrated in Fig. 4 as an example, the adopted frequency indexes in the seven time intervals are {4,7,1,6,5,2,3}. Accordingly, our method allows for a sevenelement FDA and the seven elements use the frequency offsets 4Δ f _{0}, 7Δ f _{0}, 1Δ f _{0}, 6Δ f _{0}, 5Δ f _{0}, 2Δ f _{0}, and 3Δ f _{0} with Δ f _{0} being the hopping frequency step, respectively. That is, similar to conventional FDA, all the signals are transmitted simultaneously from the Costas modulated FDA.
where α _{ p }(k), r _{ p }, and θ _{ p } are the reflection coefficient, slant range, and azimuth angle for the pth target at the kth snapshot, respectively, P is the target number, K is the snapshot number, and n(k) is the M×1 additive receiver noise vector. Note that the target reflection coefficient α _{ p }(k) may vary from shapshot to snapshot [31].
It is noticed that, like a phasedarray radar, the FDA radar has coherent transmit processing gain; however, the FDA radar directional gain depends on both the range and angle parameters, whereas the phasedarray radar directional gain depends only on the range parameter. This rangeangledependent beam provides a potential approach to suppress rangedependent interferences and noise.
which means that the FDA radar has an equivalent robustness again noise.
where \(\overline {\sigma }^{2}_{i}\) denotes the mean of \({\sigma ^{2}_{i}}\). Thus, it is expected that this FDA radar has better robustness against interferences than both conventional phasedarray radar and standard FDA radar.
4 Numerical results
It is well known that when a given delayDoppler shift results in a coincidence of N points, the ambiguity function is expected to yield a peak of approximately N/M at the corresponding delayDoppler coordinate. For the linearly frequency modulated (LFM) coding case, only delay and Doppler shifts of equal number of units, namely, τ=m t _{ b },ν=m Δ f,m=0,±1,…,±(M−1), will cause an overlap of dots, and the number of coinciding dots will be N=M−m. This hints at a diagonal ridge in the ambiguity function, along the line ν=Δ f τ/t _{ b }. What is unique for a Costas signal is that the number of coinciding dots cannot be larger than one for all but the zeroshift case, where all dots coincide (N=M). This property implies a narrow peak of the ambiguity function at the origin and low sidelobes elsewhere. If Δ f=1/τ _{1}, the exact ambiguity function values at the grid points will be either 1 or 0, according to the corresponding number of coinciding dots.
5 Conclusions
This paper proposed a ULA FDA using Costas sequence modulated frequency offsets to decouple the rangeangledependent beampattern by producing randomlike energy distribution in the transmit beampattern. Moreover, a thumbtack transmitreceive beampattern can be obtained at the receiver. In doing so, the range and angle of targets can be solely estimated through matched filtering and subspace decomposition algorithms in the receiver signal processor. Numerical results show that the proposed scheme outperforms the standard FDA in focusing the transmit energy, especially in the range dimension. This Costassequence modulated FDA radar can be exploited for LPI radar applications due to the random transmit beampattern and thumbtack transmitreceive beampattern. Such a topic is planned for our future work.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under grant 61571081, Sichuan Province Science Fund for Distinguished Young Scholars under grant 2013JQ0003, and Sichuan Technology Research and Development fund under grant 2015GZ0211.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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